Phase Gratings with Odd Symmetry for High-Resolution Lensed and Lensless Optical Sensing

ABSTRACT

A sensing device with an odd-symmetry grating projects near-field spatial modulations onto a closely spaced photodetector array. The grating includes upper and lower features that define boundaries of odd symmetry. The features include upper and lower segments of various widths on opposite sides of each boundary, the upper segments at a height sufficient to induce one half wavelength of retardation in the band of interest relative to the lower segments. The resultant interference produces the spatial modulations for capture by the array. Photographs and other image information can be extracted from the captured data.

BACKGROUND

A planar Fourier capture array (PFCA) an image sensor that does not require a lens, mirror, focal length, or moving parts. As a consequence, cameras that employ PFCAs to acquire image data can be made extraordinarily small and inexpensive.

PFCAs include angle-sensitive pixels that are sensitive to light that is sinusoidal in incident angle along the optically-sensitive axis, which can be interpreted as a measure of one component of the 2D Fourier transform of a far-away scene. Each pixel has physical characteristics that make is sensitive to a distinct component of the 2D Fourier transform of the far-away scene. Taken together, these components relate full Fourier information representative of the scene. Some applications may use the Fourier components directly, or images of the scene can be computationally reconstructed.

PFCAs exploit a near-field diffraction effect called named for Henry Fox Talbot (the “Talbot effect”). Briefly, a plane wave incident upon a periodic diffraction grating produces a repeating image of the grating at regular distances away from the grating plane. PFCAs image one of the repeating “Talbot images.” Unfortunately, it can be very difficult to manufacture devices with interlayer depths precise enough to ensure that image sensors positioned beneath a diffraction grating fall precisely at one of the regular distances that accurately reproduce a Talbot image. In standard CMOS processes, for example, interlayer thicknesses can vary by 20%. To make matters more difficult, Talbot spacing is a strong function of wavelength, making it exceedingly difficult to produce sharp Talbot images over a wavelength band of interest (e.g., the visible light spectrum).

BRIEF DESCRIPTION OF THE DRAWINGS

The detailed description is illustrated by way of example, and not by way of limitation, in the figures of the accompanying drawings and in which like reference numerals refer to similar elements and in which:

FIG. 1A is a cut-away view of a sensing device 100 with an odd-symmetry grating 105 overlying a photodetector array 110, such as a CCD (charge-coupled device) or CMOS (complementary metal-oxide-semiconductor) sensor.

FIG. 1B depicts sensor 100 of FIG. 1A simulating light incident plane 120 at an acute angle 160 to illustrate the sensitivity of curtains 140 and foci 145 to the angle of incidence.

FIG. 2 depicts a binary odd-symmetry grating 200 in accordance with one embodiment.

FIG. 3 depicts a sensing device 300 in accordance with an embodiment in which a binary, odd-symmetry phase grating 310 is formed by an interface between materials of two different refractive indices.

FIG. 4A is a plan view of a sensor 400 in accordance with another embodiment.

FIG. 4B is a three-dimensional perspective of sensor 400 of FIG. 4A, and shows how light 420 from a direction normal to the grating surface casts an interference pattern 425 on an underlying photodiode array 430.

FIGS. 5A, 5B, 5C, and 5D each depict three boundaries of odd symmetry 500 over a two-dimensional photodiode array 505.

FIG. 6 depicts three odd-symmetry gratings 600, 620, and 630, each with feature segments of different relative widths.

FIG. 7A is a cross-section of a phase grating 700 in accordance with an embodiment that uses more than two levels to produce an odd symmetry.

FIG. 7B is a cross-section of a phase grating 710 that is optically similar to phase grating 700 of FIG. 7A, but uses fewer layers.

FIG. 8 is a cross-section of a phase grating 800 that illustrates how odd symmetry can be extended to curved functions.

FIG. 9 is a plan view of a grating 900 in accordance with an embodiment in which boundaries of odd symmetry 905 extend radially from the center of the grating, and in which the widths of the feature segments widen gradually away from the center.

FIG. 10 is a plan view of a grating 1000 in accordance with an embodiment with concentric boundaries of odd symmetry 1005, and includes a cut-away view along line A-A.

FIG. 11 is a plan view of a grating 1100 in accordance with an embodiment similar to grating 900 of FIG. 9.

FIG. 12 is a plan view of a grating 1200 in accordance with another embodiment.

FIG. 13 depicts a grating 1300 in accordance with another embodiment.

FIG. 14 depicts a grating 1400 and associated photodiode array 1405.

FIG. 15 depicts a grating 1500 and associated photodiode array 1505.

FIG. 16 is a plan view of a grating 1600 in accordance with an embodiment with pentagonal boundaries of odd symmetry 1605.

FIG. 17A is a plan view of a grating 1700 in accordance with another embodiment.

FIG. 17B depicts the shapes of boundaries 1705 of FIG. 17A.

FIG. 18 depicts a two-dimensional array 1800 of gratings 1805 disposed over a photodiode array (not shown).

FIG. 19 is a flowchart 1900 detailing how an image 1905 is captured and resolved in accordance with grating 1700 of FIG. 17.

FIG. 20 depicts lithographic process for forming an image sensor 2000 in accordance with one embodiment.

FIG. 21A depicts a camera 2100 in accordance with an embodiment that includes a a lens 2105.

FIG. 21B is an example of camera 2100 with a point source 2125 imaged in focus on array 2115.

FIG. 21C is an example of camera 2100 with a point source 2140 imaged out of focus on array 2115.

FIG. 21D is an example of camera 2100 with a point source 2155 imaged more out of focus than point source 2140 in the example of FIG. 21C.

FIG. 22 is a plan view of a portion of an array of pixels 2200 illuminated with three PSFs 2205, 2210, and 2215.

FIG. 23 depicts three spiral PSFs 2300, 2305, and 2310 to illustrate how cameras in accordance with some embodiments can compensate for lens aberrations, including spherical aberration, coma, and Petzval field curvature.

FIG. 24 depicts a tessellated optical element 2400 to illustrate aspects of phase gratings in accordance with some embodiments.

FIG. 25 depicts how the rich pattern of spatial modulations within orientation chirps produced by grating 2400 of FIG. 24 facilitates improved resolution for out-of-focus PSFs.

FIGS. 26A and 26B depict tessellated gratings 2600 and 2605 in accordance with some embodiments.

FIGS. 27A and 27B depict tessellated gratings 2700 and 2705 in accordance with some embodiments.

FIG. 28 depicts a tessellated grating 2800 in accordance with one embodiment.

FIG. 29 depicts a tessellated grating 2900 in accordance with another embodiment.

FIG. 30 depicts a filter array 3000 that can be used in accordance with some embodiments to produce color images using cameras of the type detailed in FIGS. 21A-D.

FIG. 31 depicts a color channel 3100, one of four color channels for the embodiment introduced in connection with FIG. 30.

DETAILED DESCRIPTION

FIG. 1A is a cut-away view of a sensing device 100 with an odd-symmetry grating 105 overlying a photodetector array 110, such as a CCD (charge-coupled device) or CMOS (complementary metal-oxide-semiconductor) sensor. The features of grating 105 offer considerable insensitivity to the wavelength of incident light in a wavelength band of interest, and also to the manufactured distance between grating 105 and photodetector array 110. Grating 105 produces an interference pattern for capture by array 110. Photographs and other image information can then be extracted from the pattern.

Light in a wavelength band of interest—such as the visible spectrum—is incident grating 105 from a direction 115 that is normal to a transverse plane 120 of the grating 105. Dashed lines 125 highlight periodic boundaries of substantially odd symmetry. Each of these boundaries is a result of features 130 and 135 of odd symmetry, and produces a normally arranged curtain 140 of minimum intensity created by destructive phase interference between adjacent features 130 and 135. Curtains 140 are separated by foci 145, and the collection of curtains 140 and foci 145 (curtains of maximum light intensity) extend from grating 105 through the body 150 of device 100 to produce an interference pattern on photodetector array 110. In this illustration, the pattern of intensity variations evident in the foci and curtains are near-field spatial modulations that result from near-field diffraction. One photosensitive element 155 within array 110 is shaded beneath a focus 145 to serve as a reference for a subsequent discussion of the sensitivity of device 100 to the angle of incident light.

The image of FIG. 1A resulted from a simulation of a sensing device with the following parameters and assuming specific parameters. Body 150 is of fused silica, and is in contact with a conventional photodetector array 110 with photosensitive elements spaced by 2.2 um. The top of grating 105 is an air interface in this example. The relatively small segments of features 130 and 135 are about 1 um, and the relatively larger segments are about 4 um. These segments generally form transverse plane 120, which is separate from array 110 by about 25 um. Curtains 140 and foci 145 are the destructive and constructive interference patterns for 532 nm incident light.

The thickness of body 150 and lengths of the segments of features 130 and 135 were optimized for 400 nm light despite the selection of 532 nm light for the simulation. As a consequence, the tightest focus occurs about 5 um above array 110 (at the 20 um mark). The resultant curtains 140 plainly separate foci 145 well above and below the 20 um mark, however, illustrating a robust insensitivity to wavelength within the band of interest. The relatively deep and continuous penetration of curtains 140 also provides considerable manufacturing tolerance for the thickness of body 150. These advantages obtain because the near-field spatial modulations projected onto array 110 are wavelength independent over the wavelength band of interest, which means that the adjacent modulations (dark and light) do not reverse signs with changes in wavelength within the band of interest.

FIG. 1B depicts sensor 100 of FIG. 1A simulating light incident plane 120 at an acute angle 160 to illustrate the sensitivity of curtains 140 and foci 145 to the angle of incidence. Using element 155 as a reference point, we see that that the foci 145 that illuminated element 155 in FIG. 1A has considerably moved to the right in FIG. 1B. Curtains 140 and foci 145 extend at an acute angle that relates to angle 160 according to Snell's law. The separation of foci 145 by curtains 140 is maintained. Sensor 100 is thus sensitive to the angle of incidence.

FIG. 2 depicts a binary odd-symmetry grating 200 in accordance with one embodiment. Each of three boundaries of odd symmetry is indicated using a vertical, dashed line. The upper features of grating 200 are at a height sufficient to induce one half wavelength of retardation in the band of interest relative to lower features, or π radians of relative phase delay. Features 205 and 210 on either side of each boundary exhibit odd symmetry with three differently sized segments, from narrowest to widest W₁, W₂, and W₃. With this arrangement, paired segments (e.g., W₃ within features 205 and 210) induce respective phase delays that differ by approximately half a wavelength over the wavelength band of interest.

FIG. 3 depicts a sensing device 300 in accordance with an embodiment in which a binary, odd-symmetry phase grating 310 is formed by an interface between materials of two different refractive indices, a polycarbonate layer 315 and optical lanthanum dense flint glass 320 in this example. Each of four boundaries of odd symmetry 325 is indicated using a vertical, dashed line. As in the foregoing examples, the upper features of grating 310 induce phase retardations of half of one wavelength (π radians) relative to lower features. Features 330 and 335 on either side of each boundary exhibit odd symmetry. With this arrangement, paired features induce respective phase delays that differ by approximately half a wavelength over the wavelength band of interest.

These elements produce an interference pattern on an analyzer layer 327 (e.g., a conventional photodiode array) in the manner detailed in connection with FIGS. 1A and 1B. This example assumes light incident the light interface of grating 300 is normal to the transverse plane of phase grating 310, in which case light fields that enter grating 310 equidistant from a one of the boundaries of odd symmetry 325, such as at locations (−X,0) and (X,0), are out of phase at points beneath array 310 (e.g., point (0,Z)), and thus destructively interfere to produce curtains of minimum intensity (e.g., curtains 140 of FIG. 1). Neither the depth Z nor the wavelength of light over a substantial spectrum significantly influences this destructive interference. Constructive interference similarly produces foci of maximum intensity (e.g., foci 145 of FIG. 1). Both the high and low features admit light, which provides relatively high quantum efficiency relative to gratings that selectively block light.

The following discussion details phase gratings in accordance with examples described by Patrick R. Gill and David G. Stork in an upcoming paper. “Lensless Ultra-Miniature Images Using Odd-Symmetry Spiral Phase Gratings.” © 2013 Optical Society of America. In that article, Gill and Stork describe a phase grating formed by a high-n, low-dispersion substrate and a low-n, high-dispersion coating that can introduce approximately λ-independent phase shifts in all normally incident visible light. Similar gratings are discussed above. If there exist certain points p on this interface that satisfy the following symmetry in their transmission t(·) and phase retardation φ(·),

t(p+y)=t(p−y)∀y   (1)

φ(p+y)=φ(p−y)+π+2nπ ∀y, n∈I   (2)

where y is a horizontal translation transverse to the grating direction, then the grating has odd symmetry about points p, and light will interfere destructively below p, regardless of λ and depth z.

A linear odd-symmetry grating above a photosensor array could pass information from a single spatial orientation of features in the far field (transverse to the grating orientation). However, to capture information about arbitrarily oriented features of a complex scene, it is preferable to have a complete distribution of orientations in the diffractive optic. More generally, if the point-source responses (PSRs) are approximately spatially invariant, the transfer function of the imager approximates convolution with the PSR function. In such a case, the PSR should have significant power at all 2D spatial frequencies to make the inversion problem of image recovery well-conditioned.

In one example provided in Gill and Stork, gratings were numerically optimized to focus visible light onto a photodetector array 100 μm below. Optical simulations estimated the imaging performance of such a device from a 60×60 pixel array with 2.2 μm pitch 100 μm below the gratings with the sensor illuminated by a complex scene far (>>100 μm) from the sensor. The resultant photocurrent from the pixel array was unintelligible; however, the scene was reconstructed to a higher resolution than possible using a much larger diffractive imagers based on Talbot-effect angle-sensitive using Tikhonov regularization. Gill and Stork report that compressed sensing techniques could be applied to improve the reconstruction quality if the scene is known to have a compressible structure. Compressed sensing could be especially advantageous if small gaps in the Fourier transform of the PSR exist.

FIG. 4A is a plan view of a sensor 400 in accordance with another embodiment. Relatively high segments 405 and low segments 410 on either side of each of eight boundaries of odd symmetry 415 create a grating in which the widths of the segments increase with distance from the center of the sensor. For a given focal depth, light of higher frequencies tends to produce a sharper focus with narrower feature widths. Sensor 400 can therefore be optimized such that the central portion of the grating is optimized for collection of relatively higher frequency light, and the peripheral area for collection of relatively lower frequency light. This topic is detailed below in connection with other Figures.

FIG. 4B is a three-dimensional perspective of sensor 400 of FIG. 4A, and shows how light 420 from a direction normal to the grating surface casts an interference pattern 425 on an underlying photodiode array 430. Curtains and foci, as detailed previously, respectively cast shadows 435 and bright shapes 440 to be sensed by individual photosensitive elements 445 of array 430. Array 430 captures a digital representation of pattern 425.

FIGS. 5A, 5B, 5C, and 5D each depict three boundaries of odd symmetry 500 over a two-dimensional photodiode array 505. Curtains 510 cast shadows 515 on the underlying photodetectors 520, and the patterns thus created are different depending upon the angle of incident light. Array 505 can therefore sample the resultant interference pattern to obtain information as to the angle of incidence.

FIG. 6 depicts three odd-symmetry gratings 600, 620, and 630, each with feature segments of different relative widths. It can be useful to create a sensor with multiple width ratios, as shown, to compensate for manufacturing tolerances that impact the relative heights of the grating features. Assuming, for example, that grating 600 is width optimized for a manufacturing process of interest, but that the process produces a relative phase delay of 40% rather than the ideal 50% to form curtains of minimum intensity at the desired positions. To a first order the increased width of the relatively wide segments, as depicted in grating 630, can improve the distortion resulting from the erroneous phase offset. Phase offsets above 50% can be corrected for by narrowing the relatively wide segments, as depicted in grating 620. Some embodiments include a mixture of relative segment widths covering different areas of a photodiode array to accommodate manufacturing tolerances. Images associated with the gratings that provide the sharpest focus, or the sharpest focus for a wavelength of range of wavelengths, can be selected or combined to obtain the desired image data. The different gratings may also perform better for light of different wavelengths or incident angles, so selection of which gratings to use for a given image may be optimized for variables other than manufacturing tolerances.

FIG. 7A is a cross-section of a phase grating 700 in accordance with an embodiment that uses more than two levels to produce an odd symmetry. Additional levels may allow for sharper focus, but may require more complex manufacturing processes. If gratings are to be made using photolithography, for example, additional levels require additional mask steps. Paired surfaces on either side of each boundary of odd symmetry introduce respective paired phase delays that differ by approximately half a wavelength, plus an integer number of wavelengths, over the wavelength band of interest.

FIG. 7B is a cross-section of a phase grating 710 that is optically similar to phase grating 700 of FIG. 7A, but uses fewer layers. The resultant larger abrupt discontinuities 715 may introduce undesirable image artifacts or may be difficult to manufacture accurately, but the reduced number of levels may reduce manufacturing costs.

FIG. 8 is a cross-section of a phase grating 800 that illustrates how odd symmetry can be extended to curved functions.

FIG. 9 is a plan view of a grating 900 in accordance with an embodiment in which boundaries of odd symmetry 905 extend radially from the center of the grating, and in which the widths of the feature segments widen gradually away from the center. Grating 900 captures image information at sixteen discreet angles with a continuously variable set of widths. While convenient to draw grating 900 as a circle, other shapes may be used. In some embodiments, for example, collections of gratings are arrayed over a photodiode array. In such cases grids that share common boundaries (e.g., such as hexagonal, square, or triangular boundaries) make more efficient use of the underlying photodiodes.

FIG. 10 is a plan view of a grating 1000 in accordance with an embodiment with concentric boundaries of substantially odd symmetry 1005, and includes a cut-away view along line A-A. In this example the widths of the feature segments are discrete and the angles are continuous. The spacing of grating 1000 appear consistent, but may be varied to allow for sharp focus for a range of wavelengths, angles of incidence, or manufacturing variations.

FIG. 11 is a plan view of a grating 1100 in accordance with an embodiment similar to grating 900 of FIG. 9. The two halves of grating 900 provide essentially the same information. Grating 1100 adds half-circle polarization filters 1105 and 1110 with perpendicular orientations. Each half of grating 1100 thus produces image data specific to one of two polarizations, and these data can be used separately or together. More or fewer filters, with the same or different orientations, may be used in other embodiments. Different types of filters can also be used to cover all or a portion of gratings of the type described herein.

FIG. 12 is a plan view of a grating 1200 in accordance with another embodiment. Curved boundaries of odd symmetry 1205 extend radially from the center of the grating, and the widths of the feature segments widen gradually away from the center. The curvature of boundaries 1205 provide continuously varying angular information similar to what is available from grating 1000 of FIG. 10 while retaining the continuously varying spacings of grating 900 of FIG. 9.

FIG. 13 depicts a grating 1300 in accordance with another embodiment. As noted previously, different widths of the grating features provide sharper focus for different colors of light within the wavelength band of interest. Grating 1300 has the same radial symmetry of grating 900 of FIG. 9, but those areas for which the spacing is optimized for blue, green, and red light are provided with filters to admit their respective wavelengths. Omitting wavelengths that provide a blurred interference pattern on the underlying analyzer can improve image sharpness, and can allow more accurate reconstruction of color image data. Grating 1300 is bounded by an opaque mask 1305 that defines the limit of the aperture.

FIG. 14 depicts a grating 1400 and associated photodiode array 1405. Grating 1400 has parallel odd-symmetry boundaries 1410, which may have features of the same or different widths, or of varying widths along one or more boundaries. Parallel boundaries with the requisite diversity of widths and spacings to sample a sufficient number of spatial frequencies can image one-dimensional images, e.g., barcodes. Array 1405 is shown alongside, rather than below, grating 1400 to highlight the angle θ_(A) between the direction of boundaries 1410 and the columns of photosensitive elements in array 1405. Angle θ_(A) creates more diversity of measurements because the linear shadow covers different percentages of pixels in different rows. In one embodiment angle θ_(A) is selected so that the top of each boundary is offset from the bottom by about one pixel of array 1405.

FIG. 15 depicts a grating 1500 and associated photodiode array 1505. Grating 1500 has parallel, right-angled boundaries 1510, which may have features of the same or different widths, or of varying widths along one or more boundaries. Parallel boundaries with the requisite diversity of widths and spacings along two dimensions to sample a sufficient number of spatial frequencies can image e.g. point sources, such as to identify the position of the sun. Angle θ_(A) can be introduced for the reasons presented above in connection with FIG. 14.

FIG. 16 is a plan view of a grating 1600 in accordance with an embodiment with pentagonal boundaries of odd symmetry 1605. In this example the widths of the feature segments are discrete, but they can vary along one or more boundaries in other embodiments. Straight boundaries may be advantageous over curved ones because line segments can more easily provide precise odd symmetry.

Grating 1600 provides information at five different orientations. Other boundary shapes, such as other polygons, are used in other embodiments. In general, polygons with odd numbers of sides provide greater orientation diversity than polygons with a similar but even number of sides (e.g., a pentagon provides more orientation diversity than a square or a hexagon).

FIG. 17A is a plan view of a grating 1700 in accordance with another embodiment. Recalling that relatively narrow (wide) segment spacing works better for relatively high (low) frequencies, feature spacing increases along odd-symmetry boundaries (between dark and light regions) with distance from the center. Curved boundaries of odd symmetry 1705 extend radially from the center of the grating to the periphery, radiating out between the dark (elevated) and light (recessed) arms near the center. The curved boundaries are obscured by grating features in FIG. 17A, so the shapes of boundaries 1705 are depicted in FIG. 17B for ease of review.

The segment widths do not continue to increase with radius, as there is a maximum desired width for a given wavelength band of interest (e.g., the widest may correspond to the lowest frequency of visible red light). The features that define boundaries 1705 therefore exhibit discontinuities as they extend toward the periphery of grating 1700. In this example, grating 1700 has three discrete areas each tuned to a subset or all of the wavelengths in the band of interest.

FIG. 18 depicts a two-dimensional array 1800 of gratings 1805 disposed over a photodiode array (not shown). Each of gratings 1805 is identical, but any number of parameters, many of which are discussed previously, can be varied within and among gratings 1805. For example, different shapes and types of gratings can be used to create and image different types of interference patterns that can be combined or used separately to obtain some desired result. The decision to consider all or a specific subset of information generated by one or more of the constituent gratings can be done once, such as at time of manufacture to accommodate process variations, or can be done dynamically to highlight different aspects of a scene. Emphasizing aspects of different patterns can be used, for example, to highlight light of different polarizations, wavelengths, or angles of incidence.

Spaced gratings facing the same direction, particularly when their characteristics are well matched, can be used to sense moving objects. Assuming matched gratings with a fixed separation receiving light from the same scene, the difference between the photocurrents of the respective analyzer layers is sensitive only to objects relatively close to the pair. Further, the time derivative of this difference is sensitive to nearby, moving objects, and is insensitive to relatively distant moving or stationary objects.

FIG. 19 is a flowchart 1900 detailing how an image 1905 is captured and resolved in accordance with grating 1700 of FIG. 17. First, an image 1910 is presented such that light from image 1910 is incident grating 1700. The incident light passes through phase grating 1700 to produce an intensity pattern 1920 on an underlying two-dimensional array of photosensors (not shown), which captures the pattern (1915). The captured pattern 1920 may appear unintelligible to a human; however, because grating 1700 has sharp features in its point-spread function (PSF), the pattern contains rich information about the image.

The PSF of grating 1700, possibly in combination with the underlying array, is known from a prior calibration or high-fidelity simulation. The way in which the PSF varies as a function of incident angle and color may also be similarly determined. This information is represented by a response 1930. A mathematical conversion based on this response can thus be used to reconstruct image 1910 from pattern 1920.

To recover the original image, responses 1920 and 1930 are combined to form an inverse problem (1925), which is solved (1935) to recover a version 1940 of the original image. One embodiment employs the well-known Tikhonov regularized inversion technique to accomplish steps 1925 and 1935. Take as a starting point a) detailed knowledge of the PSF of grating 1700, b) knowledge of the noise level of the system under current illumination conditions, and c) the specific readings observed for this image (pattern 1920). Express the unknown image as an N×1 vector x, where N is the total number of pixels one wishes to reconstruct. Express the readings from the photosensor as an M×1 vector y, where M is the total number of photosensors in the array. Express detailed knowledge of the PSF as an M×N matrix A such that for any image x, the formula yielding expected observed signal y under x is y=Ax, called the “forward equation.”

To reconstruct an image, it suffices to solve the forward equation with a known measurement vector y for an unknown image x as follows. Multiply both sides of the forward equation by A^(T) (the transpose of A) to obtain A^(T) y=A^(T) Ax. The matrix A^(T) A is square and in principle could be directly inverted to recover x; however usually this inversion is poorly conditioned when noise is present and when not all eigenvectors of A^(T) A have equally large associated eigenvalues. Thus in practice, Tikhonov regularization (as follows) usually delivers preferable results.

Next, select a regularization parameter λ>0 based on the noise level at the current illumination conditions. Finally, invert the matrix (A^(T) A+λI) (where I is the identity matrix), assume (A^(T) A+λI)≈(A^(T) A) and multiply on the left of the preceding equation to obtain x≈(A^(T) A+λI)³¹ ¹ A^(T) y. Therefore, for a given regularization parameter λ, the image recovered through Tikhonov regularization is a linear combination of the readings from the photosensor. If the PSF is sufficiently spatially invariant to the extent that its spatial dependence can be neglected, these computations can be done in the Fourier domain, allowing for much faster numerics.

Another embodiment recovers the matrix x using compressed sensing. If the scene is expected to be sparse in some basis (such as a wavelet transform W for natural images), the following methodology can be used. We can recover the sparse scene components z where x=Wz by finding the z that minimizes the following cost function: ½r^(T)r+λf(z), where r is the residual (y−AWz), λ>0 is a regularization parameter (different from that used in Tikhonov regularization, but also noise-dependent), and f(z) is a function penalizing non-sparse z. If f(z) is a convex function of z such as the L₁ norm, this optimization problem can be solved efficiently using convex optimization techniques. The penalty function f(z) can also take on other forms, including terms penalizing total variation in the reconstructed image x or other prior scene knowledge.

Some of the chief advantages of compressed sensing over linear approaches such as Tikhonov regularization are that the former allow more prior information about the expected scene structure to help shape the final image. Further, if A^(T) A does not have full rank or cannot measure certain aspects of the scene (for example, due to some near-zero regions of the 2D Fourier transform of the PSF), using compressed sensing sometimes overcomes these limitations given correct prior information about the expected images.

The foregoing Tikhonov and compressed-sensing techniques can include iterative methods to reduce problem complexity. For example, Richardson-Lucy deconvolution can iteratively approximate Tikhonov regularized inversion and iterated wavelet thresholding can be a numerically-efficient way to converge to a compressed-sensing-like solution.

In some embodiments the purpose of the sensor is not to reconstruct an image, but to perform some optical sensing task. In such cases the vector x may represent the sought measurement rather than the field of image pixels, and the forward transform A can be appropriately modified.

FIG. 20 depicts a lithographic process for forming an image sensor 2000 in accordance with one embodiment. First, a wafer 2005 of material that is transparent over the wavelength band of interest is patterned with a mask 2010 that defines the relatively high features of what will become an odd-symmetry grating surface of the type detailed herein. Next, the exposed surface of wafer 2005 is etched to create recessed regions 2015. Mask 2010 is then removed. Finally, wafer 2005, now comprising a grating, is bonded to a photodiode array 2025. Photolithographic and wafer-bonding processes are well known to those of skill in the art, so a detailed discussion is omitted.

FIG. 21A depicts a camera 2100 in accordance with an embodiment in which a converging optical element, in this case a lens 2105 (although a single-element lens is shown for simplicity of illustration, generally the optical element can comprise one or more refractive, diffractive, and/or reflective elements), is used in conjunction with a phase grating element, grating 2110, disposed in the path between the optical element and a dense photodetector array 2115 to form images thereon. A scene incident the front side of lens 2105 is projected through grating 2110 and onto array 2115. Grating 2110 induces spatial modulations in the incoming light and passes the resulting interference pattern to array 2115, which captures a digital representation of the spatial modulations. An integrated processor 2120 electrically coupled to array 2115 computes an image of the scene from the digital representation. The processor is shown also physically coupled to array 2115, but the processor can be located elsewhere in other embodiments.

Lens 2105 defines a front focal point FFP and a rear focal point RFP, and is spaced from grating 2110 by a distance less than the image-plane distance D between lens 2105 and focal point RFP. Array 2115 is on the opposite side of focal point RFP from grating 2110 in this example. Grating 2110 may be an odd-symmetry grating that has properties detailed above in connection with the preceding figures. In other embodiments (such as an embodiment primarily operating in a macro mode) the focal length of lens 2105, defined for objects at infinity, may be closer to lens 2105 than to grating 2110, or may move over a range that encompasses such relative positioning.

Surface features of grating 2110 are separated from array 2115 by a distance X. Though shown as separate structures for ease of illustration, grating 2110 can be integrated with or attached to array 2115. Distance X in camera 2100 is, in this example, no more than 400 times a longest wavelength of interest λ_(max) in the medium(s) between the surface features of grating 2110 and array 2115 (X≦400λ_(max)). For example, a camera in which λ_(max) is 0.5 microns may have a spacing X between the features of grating 2110 and the surface of array 2115 of up to 200 microns.

FIG. 21B is an example of camera 2100 with a point source 2125, represented by tip of an arrow, that is imaged in focus on array 2115. Grating 2110 is out of the focal plane, so the light from lens 2105 presents a blur-spot PSF 2130 to grating 2110. (As in other examples used herein, the area occupied by PSF 2130 refers to the area of the central lobe.) Grating 2110 produces an interference pattern from function 2130, but the illumination boundaries of the pattern are not evident in the tightly focused, diffraction-limited spot 2135 on array 2115. Objects at the range and position of point source 2125 are tightly focused (field curvature and other aberrations may change the best focus range for other positions), and are nominally imaged at the full resolution of array 2115, assuming lens 2105 is capable of such resolution.

FIG. 21C is an example of camera 2100 with a point source 2140 that is imaged out of focus on array 2115. As in the prior example, the light from lens 2105 presents a blur-spot PSF 2145 to grating 2110, and grating 2110 produces a pattern of spatial modulations. Because point source 2140 is imaged out of focus, however, the area of PSF 2150 at array 2115 is greater than in the example of FIG. 21B, and illumination transitions/substructure within the pattern area are evident. In camera 2100, these illumination transitions are near-field spatial modulations induced by features of grating 2110. The resultant spiral pattern of PSF 2150 is preferably an invertible orientation chirp. As used herein, an “orientation chirp” is a pattern of spatial modulations that cover ranges of spatial frequencies and orientations sufficient to recover an image at a desired resolution.

FIG. 21D is an example of camera 2100 with a point source 2155 that is imaged more out of focus than point source 2140 in the example of FIG. 21C. Light from lens 2105 presents a blur-spot PSF 2160 that is still greater than PSF 2145, and a resultant invertible PSF 2165 on array 2115 is similarly larger than PSF 2150. Although not shown, imaging a point source at the FFP of FIG. 21A produces an invertible PSF including orientation chirp features. Two point sources, one in front of and one behind point 2125 but along the same optical axis, may produce similar-sized orientation chirps. Due to aberrations in the lens system, however, the chirps may differ—such differing characteristics may be used to resolve range, as detailed further below.

FIGS. 21A-D illustrate the general point that the pattern area and the richness of the accompanying spatial modulations on array 2115 are a function of focus, the duller the focus the greater the area and the better resolved the spatial modulations. Point sources farther away from lens 2105 than point source 2125 of FIG. 21A produce ever larger PSFs on the array as they move away from (or towards) lens 2105.

The PSF for an out-of-focus point source is a scaled version of an orientation chirp from grating 2110, where the diameter of the orientation chirp is proportional to defocus of the point source. The observations at the sensor plane (the surface of array 2115) will therefore be the in and out-of-focus imaged points, each convolved with the orientation chirp at a chirp phase dependent upon the position the light ray bundle received from that point, scaled according to an out-of-focus parameter, and spatially superimposed with like contributions from other imaged points. Camera 2100 can recover relatively high-resolution images of out-of-focus objects because this convolution is computationally invertible for the majority of common image capture situations. In this context, “computationally invertible” means that image data can be recovered to a specified degree of precision using e.g. inverse, pseudoinverse, and compressed-sensing transformations. A PSF is computationally invertible, for example, if its 2D Fourier transform is “complete,” or has substantial amplitude at all spatial frequencies required to recover an image at a specified resolution.

Not all spiral PSFs are complete. For example, Archimedean spirals have regularly spaced arms whose Fourier transforms have peaks at the reciprocal of the inter-arm period and nulls between these peaks. In contrast, the spiral PSF 1930 of FIG. 19 has few, unevenly spaced arms that are sharply bounded and sweep through all orientations, so it has significant Fourier power at all spatial frequencies and is complete. Due to this completeness, accurate deconvolution is relatively well-conditioned, so undoing the effect of the PSF is relatively straightforward. Regardless of whether computations are performed in the Fourier domain or the spatial domain, deconvolution works well if the Fourier transform of the PSF has no zeros. In the case that a point source causes a blur spot 2160 that is not concentric with a spiral, the resulting PSF will contain a spatially wrapped version of the PSF. Spatially wrapping the spiral does not substantially affect its completeness.

Camera 2100 can measure light intensity from photodetector array 2115 without first needing to focus (although some embodiments can focus manually or automatically). Data captured by array 2115 includes orientation chirps with Fourier-component strengths that vary with depth (see FIGS. 21B-D). The Fourier transform of the local observations will be the product of the imaged object's Fourier transform and the depth-dependent Fourier transform of the orientation chirp. By searching for the depth-specific kernel that best matches this product for each imaged point, scene depth can be determined, assuming the scene has some texture, as detailed below.

The depth d of a local scene patch x can be inferred from readings y through Bayesian estimation as follows. First, a likelihood p(y|d) of each depth can be computed by a further Bayesian estimation based on knowledge that the Fourier transform of y is the product of the Fourier transforms of x and the depth-dependent PSF, and with knowledge of typical power spectra of photographed objects. Next, this likelihood p(y|d) is weighted by a Bayesian prior on the known distribution of depths and depth changes in a scene to arrive at a posterior probability of p(d|x) for depth at each point in the scene. Bayesian estimation of the depth map of a scene based on depth and depth change priors, as well as point-wise estimates of depth associated with corresponding certainty (indicated by the height of peaks in the likelihood p(y|d)) is a technique known to those skilled in the art, and will not be further discussed here. In this application, knowledge of the true depth map is important for accurate image recovery (to be described shortly) precisely for those images that have significant Fourier power in spatial frequencies that interact with the Fourier transform of the PSF. Thus, accurate depth maps are possible where the scene has fine texture, and where scene patches lack this texture convolution with the PSF does not degrade image quality in the scene.

Next, the Fourier transforms are deconvolved in image space or the Fourier domain; the problem scale will dictate which of these is faster. The deconvolution kernel can also be made to vary with light level for a Weiner-optimal reconstruction (although humans tend to prefer overly-sharpened images; this sharpening filter can be incorporated with the deconvolution filter to save an additional step).

The result of selecting the correct filter followed by deconvolution is a depth map and a reconstruction of the original image. If the orientation chirp is Fourier-complete, the reconstructed image can resolve the same number of pixels as array 2115. This is unlike most plenoptic cameras, and is made possible by the fact that each pixel reading contributes useful information to the deconvolution problem. In the case where a PSF's high-frequency components are small, processor 2120 may smooth the highest spatial frequencies to avoid adding too much noise. In low-light conditions, camera 2100 may lose e.g. a factor of two in resolved pixels due to this effect; this represents an improvement over existing plenoptic cameras, whose pixel efficiency may be as low as 4%. For well-formed orientation chirps according to an embodiment and general imaging conditions, PSFs with a central lobe diameter up to six photodetector pitches should be invertible to recover image features with a spatial frequency up to at least 0.25 cycles per photodetector (Nyquist frequency being 0.5 cycles per photodetector pitch in the major dimensions of the photodetector array). Such performance depends in part on the lens element having a sufficient modulation transfer function at the relevant spatial frequencies.

FIG. 22 is a plan view of a portion of an array of pixels 2200 illuminated with three PSFs 2205, 2210, and 2215. PSF 2205 is an orientation chirp representing a sharply focused point source; illumination substructure cannot be resolved given the pitch of array 2200. If all points of a given scene are in focus, image resolution is primarily a function of array pitch, or of array pitch and the diameter of a diffraction-limited spot.

PSF 2210 is an orientation chip representing a poorly focused point source; spatial modulations appear as spiral arms of a computationally rich PSF that can be resolved by array 2200 to locate the corresponding point source in the image. Finally, PSF 2215 represents a point source whose focus is between those of PSFs 2205 and 2215; spatial modulations can again be resolved to locate the corresponding point source in the image.

For both PSF 2210 and 2215, the resolution of the image is limited by the larger of the pitch and the spacing of the separation between arms of the PSF spiral. In this illustration, the three point sources are easily located in the two dimensions of array 2200. Further, the three disparate pattern areas of the three PSFs provide a measure of distance in a dimension normal to array 2200. Cameras like camera 2100 of FIGS. 21A-D can therefore provide extended depths of field, focused images for out-of-focus objects, and measures of distance from image data.

FIG. 23 depicts three spiral PSFs 2300, 2305, and 2310 to illustrate how cameras in accordance with some embodiments can compensate for lens aberrations, including spherical aberration, coma, and Petzval field curvature. Such compensation can simplify primary lens design and allow an increase in aperture without sacrificing image quality.

Spherical aberration is the condition whereby the focal length of a given annulus of a lens varies linearly with the annulus' radius. In the configuration of FIG. 21, this condition may influence the shapes of orientation chirps on the array. PSF 2300 of FIG. 23 is a hypothetical ideal chirp, the result of a perfect lens. PSF 2305 shows a type of chirp distortion that may result from a lens with spherical aberration. As compared with PSF 2300, PSF 2305 has relatively linear arms near the center. So long as the orientation chirp is complete (invertible to recover the image data), imaging performance will not be degraded. Even if not complete, imaging performance may be acceptable if the orientation chirp is sufficiently invertible to recover images to a desired resolution.

A lens has coma if light passing through different annuli centered on the lens forms annuli on the image sensor whose center varies with annulus radius. As shown in PSF 2310, coma produces an elongated and distorted, but complete spiral. Petzval field curvature is the aberration whereby the lens' focal surface is not planar. As with spherical aberration, coma, Petzval field curvature, and other aberrations can be undone if the orientation chip is sufficiently complete.

Lens aberrations can be beneficial in some embodiments. A PSF out-of-focus to one side of the image plane can cover a pattern area of the same size as a PSF out-of-focus to the other side of the image plane. If two such PSFs are identical, then the camera may not be able to distinguish between them. Lens aberrations can render such PSFs distinguishable, however, such as by producing opposite asymmetries, and can therefore allow cameras in accordance with some embodiments to better distinguish point sources along the axis of incidence.

FIG. 24 depicts a tessellated optical element 2400 to illustrate aspects of phase gratings in accordance with some embodiments. Element 2400 is tessellated with spiral ensembles 2410 of sub-elements 2405—depicted as curvilinear boundaries—that are contiguous across tessellation borders (the hexagonal borders are for illustration, and do not represent physical structure in this example). The sub-elements of each ensemble are arranged such that light converged by element 2400 from a point source and passing through one of ensembles 2410 forms a PSF with spatial modulations representative of the ensemble. In one aspect, the tessellated optical element further converges what would otherwise strike a sensor array as a blurry PSF into a PSF that, while of similar size to the hypothetical PSF, contains high-frequency substructure.

Returning for a moment to the example of FIG. 21D, the blur spot PSF 2160 is assumed to be centered on an ensemble of spiral features to produce the spiral PSF 2165. This is a somewhat special case. Point sources at the same distance from the camera yet in general position will have a PSF containing all sub-elements 2405 of at least one ensemble 2410 collected from neighboring ensembles, with some of them spatially wrapped around. In the example of FIG. 24, a PSF outline 2415 represents the area of a central lobe that is off center with respect to any of the sub-gratings 2410, but that nevertheless covers enough grating features 2405 to produce an invertible orientation chirp. In general, it is beneficial that the wrapping of spatial features between ensembles 2410 not substantially alter the magnitude of the components of the Fourier transform of the resultant orientation chirp. A circle like outline 2415, of sufficient area to encompass one of ensembles 2410, can be swept along a path between neighboring ensembles while, for all intermediate circle positions along the swept path, the swept circle contains optical sub-elements arranged at all the orientations contained in the circle at the start of the path (e.g., all positions produce similar spectra, but with shifting phase).

FIG. 25 depicts how the rich pattern of spatial modulations within orientation chirps produced by grating 2400 of FIG. 24 facilitates improved resolution for out-of-focus PSFs. As in other examples, the preceding digit or digits of each element name indicate the figure in which the element was introduced. Using this convention, elements 24## and 25## refer to features depicted in FIGS. 24 and 25, respectively.

In the top row of FIG. 25, light rays from a point source 2500 pass through a lens (not shown) and onto tessellated grating 2400 of FIG. 24 over the area 2505 encompassed by outline 2415 as blurred PSF 2510. The grating creates orientation chirp 2515, which includes a rich set of spatial modulations as a consequence of the sub-elements 2405 within the area of outline 2415. Chirp 2515 is not a spiral because PSF 2510 is not centered on an ensemble (the pattern of chirp 2515 is shown to match the pattern of sub-elements 2405 within the PSF outline 2415). However, the spatial modulations of chirp 2515 are sufficient that chirp 2515 is invertible.

The second row of FIG. 25 is similar to the first, but includes light rays from adjacent point sources 2520 that illuminate overlapping areas 2525 to produce a pair of overlapping, blurred PSFs 2530. The grating creates a discernible pair of orientation chirps 2535 and 2540, the locations of which can be computationally inverted to a higher resolution than could the smeared PSFs 2530. Chirps 2535 and 2540, shown separately to the right, are slightly different from one another because each PSF 2530 impinges upon a slightly different area of the grating.

The third row of FIG. 25 shows a constellation of nine point sources 2542 that illuminate an area 2545 on the grating with overlapping, blurred PSFs 2550, and the resultant nine orientation chirps collectively labeled 2555. As in the last example, the locations of the point sources corresponding to the orientation chirps 2555 can be resolved with far greater precision than could be accomplished using the PSFs 2550.

FIGS. 26A and 26B depict tessellated gratings 2600 and 2605 in accordance with some embodiments. Grating 2600 is depicted using boundaries between high and low features, whereas grating 2605 depicts relatively high and low features in black and white, respectively.

FIGS. 27A and 27B depict tessellated gratings 2700 and 2705 in accordance with some embodiments. Grating 2700 is depicted using boundaries between high and low features, whereas grating 2705 depicts relatively high and low features in black and white, respectively.

FIG. 28 depicts a tessellated grating 2800 in accordance with one embodiment. Grating 2800 depicts relatively high and low features in black and white, respectively.

FIG. 29 depicts a tessellated grating 2900 in accordance with another embodiment. Grating 2900 depicts relatively high and low features in black and white, respectively.

FIG. 30 depicts a filter array 3000 that can be used in accordance with some embodiments to produce color images using cameras of the type detailed in FIGS. 21A-D. Filter array 3000 includes four color filters, a red filter 3005, two green filters 3010 and 3015, and a blue filter 3020. Each filter is associated with what amounts to an instance of a camera like camera 2100 of FIG. 21A that acts as one of four color channels for the overall camera. For each camera like camera 2100, the wavelength band of interest is limited to the wavelengths passed by the color filter in the optical path.

FIG. 31 depicts a color channel 3100, one of four color channels for the embodiment introduced in connection with FIG. 30. Channel 3100 is similar to camera 2100 of FIGS. 21A-D, so a detailed discussion is omitted. Briefly, channel 3100 includes a color filter, a lens 3105 whose optical properties should be tuned for the light frequencies admitted by the color filter, a grating 3110, and a photodetector array 3115. The red filter 3005 of FIG. 30 is inserted somewhere in the optical path and covering the entire field of view, in this case between lens 3105 and grating 3110. Characteristics of channel 3100, such as the focal length of lens 3105, the spacing X between grating 3110 and array 3115, the spatial frequency range of grating 3110, the depth of grating features and composition of grating 3110, and the geometry of the grating sub-elements may be optimized for the selected color. An image processor (not shown) can combine information collected by the color channels to recover a color image.

The color channels can have fixed lenses with distinct focal lengths, or can have the same lens but distinct spacing between the lens and grating. In cameras with multiple channels of diverse focal lengths, the channel or channels presenting the sharpest focus may be used to capture high-resolution scene information, while the other, relatively defocused channels, can provide color information. Techniques to “paint on” relatively low resolution color information onto a high-resolution image are well known to those of skill in the art.

Cameras with multiple channels, whether for the same or different wavelength bands of interest, provide measures of parallax that can be combined with other information derived from orientation chirps to make a depth map of a scene. Also advantageous, information from multiple channels can be used to disambiguate depth in the case where the object exhibits deceptive structure. For example, a scene with in-focus spiral patterns may appear defocussed to a spiral ensemble. In such cases a cross check to one or more additional channels can resolve ambiguity by selecting which “reality” is most consistent with the image data provided by the disparate channels.

While the subject matter has been described in connection with specific embodiments, other embodiments are also envisioned. For example; while each grating detailed previously may be used in connection with photoreceptors to collect incident light, gratings in accordance with these and other embodiments can be used more generally in imaging devices that project images from photo-emitters rather than or in addition to sensing them; cameras described as using lenses could also employ other types of optical elements (e.g., mirrors); the wavelength band of interest can be broader or narrower than the visible spectrum, may be wholly or partially outside the visible spectrum, and may be discontinuous; and cameras and gratings detailed herein can be adapted for use in multi-aperture or programmable-aperture applications. The wavelength band of interest is the visible spectrum in these examples. Other variations will be evident to those of skill in the art. Therefore, the spirit and scope of the appended claims should not be limited to the foregoing description. Only those claims specifically reciting “means for” or “step for” should be construed in the manner required under the sixth paragraph of 35 U.S.C. §112. 

1. (canceled)
 2. An imaging device to sense patterns of incident light over a wavelength band of interest, the imaging device comprising: a photodetector array defining a focal plane; and a phase grating defining a transverse plane to receive the incident light, the phase grating including: a pair of adjacent first grating segments, of a first segment width W₁, defining a boundary of odd symmetry extending in the transverse plane, the first grating segments inducing destructive interference below the boundary of odd symmetry and at the focal plane for the incident light within the wavelength band of interest; a pair of second grating segments, one on either side of the pair of first grating segments, of a second segment width W₂ greater than the first segment width, the second grating segments inducing destructive interference below the boundary of odd symmetry and at the focal plane for the incident light within the wavelength band of interest; and a pair of third grating segments, one on either side of the pair of second grating segments, of a third segment width W₃, the third grating segments inducing destructive interference below the boundary of odd symmetry and at the focal plane for the incident light within the wavelength band of interest.
 3. The imaging device of claim 2, wherein the third segment width W₃ is greater than the second segment width.
 4. The imaging device of claim 2, further comprising additional pairs of the first grating segments defining additional boundaries of odd symmetry.
 5. The imaging device of claim 4, further comprising additional pairs of the second grating segments bounding the additional pairs of the first grating segments.
 6. The imaging device of claim 5, wherein each of the third grating segments is between two of the boundaries of odd symmetry.
 7. The imaging device of claim 6, wherein each of the third grating segments is centered between the two of the boundaries of odd symmetry.
 8. The imaging device of claim 2, wherein one of each of the pairs of grating segments retards the incident light by half of a wavelength of light within the wavelength band of interest relative to the other one of the pair.
 9. The imaging device of claim 2, wherein the boundaries of odd symmetry are curved.
 10. The imaging device of claim 2, wherein the first width W₁ varies in the transverse plane.
 11. The imaging device of claim 10, wherein the boundaries of odd symmetry intersect at an intersection, and the first width W₁ increases along the boundaries with a distance from the intersection.
 12. The imaging device of claim 11, wherein the intersection is at a center of the grating.
 13. The imaging device of claim 11, wherein the first width W₁ increases to a maximum first width, narrows at a discontinuity where its retardation changes by one half the wavelength, plus an integer number of the wavelength, and increases from the discontinuity as a function of the distance from the intersection.
 14. The imaging device of claim 2, wherein each of the second grating segments is immediately adjacent one of the second grating segments and one of the third grating segments.
 15. The imaging device of claim 2, wherein the phase grating includes a body in contact with the photodetector array.
 16. The imaging device of claim 2, wherein the focal plane is greater than twenty microns from the transverse plane.
 17. An imaging device comprising: an array of photo-elements operable to communicate sets of light rays incident the array and having an average wavelength in a wavelength band of interest; and a transmissive phase grating defining a transverse plane opposite the array of photo-elements, the phase grating spaced from the array and having boundaries of odd symmetry, each boundary defined by: a first phase-grating feature extending along the respective boundary in the transverse plane, each first phase-grating feature including a first segment adjacent the boundary and of a first width W₁ in a dimension perpendicular to the boundary and within the plane, a second segment adjacent the first segment and opposite the boundary, the second segment of a second width W₂ different from the first width in the dimension, and a third segment adjacent the second segment and opposite the boundary, the third segment of a third width W₃ in the dimension; and a second phase-grating feature extending along the respective boundary in the transverse plane, each second phase-grating feature including a first segment adjacent the boundary and of the first width W₁ in the dimension perpendicular to the boundary and within the plane, a second segment adjacent the first segment and opposite the boundary, the second segment of the second width W₂ in the dimension, and a third segment adjacent the second segment and opposite the boundary, the third segment of the third width W₃ in the dimension.
 18. The device of claim 17, wherein the first segment of the first phase-grating feature induces retardation of one half the wavelength, plus an integer number of the wavelength, relative to the first segment of the second phase-grating feature, and the second segment of the second phase-grating feature induces retardation of one half the wavelength, plus an integer number of the wavelength, relative to the second segment of the first phase-grating feature.
 19. The device of claim 17, wherein the third width W₃ is wider than the second width W₂.
 20. The device of claim 17, wherein the boundaries are curved.
 21. The device of claim 17, wherein the boundaries diverge.
 22. The device of claim 17, wherein the boundaries are discontinuous.
 23. An imaging device comprising: a photodetector array defining a focal plane; and a phase grating defining a transverse plane to receive incident light over a wavelength band of interest, the phase grating including: first segments to induce first phase delays of the incident light; second segments between the first segments, the second segments to induce second phase delays of the incident light, the second phase delays differing from the first phase delays by half a wavelength over the wavelength band of interest; wherein: adjacent pairs of the first and second segments of a first width W₁ define lines of symmetry; pairs of the first segments of the first width W₁ are separated by ones of the second segments of a second width W₂ greater than the first width W₁; and pairs of the second segments of the first width W₁ are separated by ones of the first segments of the second width W_(2.)
 24. The imaging device of claim 23, wherein each of the ones of the first and second segments of the second width W₂ is adjacent one of the first and second segments of the first width W₁.
 25. The imaging device of claim 23, wherein each of the ones of the first and second segments of the second width W₂ is adjacent two of the first and second segments of the first width W₁.
 26. The imaging device of claim 23, wherein the lines of symmetry comprise boundaries of odd symmetry.
 27. The imaging device of claim 23, wherein a first sum of the widths of the first segments between adjacent first and second ones of the lines of symmetry equals a second sum of the widths of the second segments between the second of the lines of symmetry and a third of the lines of symmetry adjacent the second of the lines of symmetry.
 28. The imaging device of claim 23, wherein the lines of symmetry are curved.
 29. The imaging device of claim 23, wherein the first width W₁ varies in the transverse plane.
 30. The imaging device of claim 29, wherein the lines intersect at an intersection, and the first width W₁ increases along the lines with a distance from the intersection.
 31. The imaging device of claim 23, further comprising an optical element to focus the incident light onto the phase grating.
 32. The imaging device of claim 23, wherein adjacent pairs of the first and second segments of different widths define lines of a symmetry, each line of a symmetry less than forty-five degrees from a nearest line of symmetry. 